A two-stage flow shop batch-scheduling problem with the option of using Not-All-Machines

The decision whether to use all the available machines in the shop becomes very relevant when the capacity exceeds the demand. In such cases, it might be optimal to use only a subset of the machines. We study this option in a two-stage flowshop environment. Jobs are assumed to be identical, and are processed in batches, where a machine-dependent setup time is required when starting a new batch. The objective function is minimum makespan. We introduce an exact efficient dynamic programming algorithm, which is shown numerically to be able to solve medium size instances in very reasonable time. For the solution of large instances, we propose an asymptotically optimal heuristic procedure and a lower bound on the makespan value, which produce extremely small optimality gaps.